Properties

 Label 2352.4.a.cp.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$13.2349$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -18.9580 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -18.9580 q^{5} +9.00000 q^{9} -54.7308 q^{11} -62.0173 q^{13} -56.8741 q^{15} -122.439 q^{17} -12.5058 q^{19} +74.4391 q^{23} +234.406 q^{25} +27.0000 q^{27} -232.572 q^{29} +10.3677 q^{31} -164.192 q^{33} -245.987 q^{37} -186.052 q^{39} -238.653 q^{41} +92.9718 q^{43} -170.622 q^{45} -485.645 q^{47} -367.317 q^{51} -378.557 q^{53} +1037.59 q^{55} -37.5173 q^{57} -182.784 q^{59} -396.470 q^{61} +1175.73 q^{65} +261.239 q^{67} +223.317 q^{69} +874.523 q^{71} -152.406 q^{73} +703.219 q^{75} -573.357 q^{79} +81.0000 q^{81} +317.754 q^{83} +2321.20 q^{85} -697.717 q^{87} +95.0160 q^{89} +31.1032 q^{93} +237.084 q^{95} +1608.78 q^{97} -492.577 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} - 4q^{5} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} - 4q^{5} + 36q^{9} - 14q^{11} - 22q^{13} - 12q^{15} - 96q^{17} - 26q^{19} - 96q^{23} + 110q^{25} + 108q^{27} - 76q^{29} + 238q^{31} - 42q^{33} + 562q^{37} - 66q^{39} - 428q^{41} + 258q^{43} - 36q^{45} - 80q^{47} - 288q^{51} + 1476q^{55} - 78q^{57} + 262q^{59} + 276q^{61} + 2196q^{65} - 150q^{67} - 288q^{69} + 848q^{71} + 218q^{73} + 330q^{75} - 1762q^{79} + 324q^{81} + 3450q^{83} + 1452q^{85} - 228q^{87} + 344q^{89} + 714q^{93} - 2004q^{95} + 622q^{97} - 126q^{99} + O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −18.9580 −1.69566 −0.847828 0.530271i $$-0.822090\pi$$
−0.847828 + 0.530271i $$0.822090\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −54.7308 −1.50018 −0.750089 0.661337i $$-0.769989\pi$$
−0.750089 + 0.661337i $$0.769989\pi$$
$$12$$ 0 0
$$13$$ −62.0173 −1.32312 −0.661558 0.749894i $$-0.730104\pi$$
−0.661558 + 0.749894i $$0.730104\pi$$
$$14$$ 0 0
$$15$$ −56.8741 −0.978988
$$16$$ 0 0
$$17$$ −122.439 −1.74681 −0.873407 0.486991i $$-0.838095\pi$$
−0.873407 + 0.486991i $$0.838095\pi$$
$$18$$ 0 0
$$19$$ −12.5058 −0.151001 −0.0755004 0.997146i $$-0.524055\pi$$
−0.0755004 + 0.997146i $$0.524055\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 74.4391 0.674853 0.337427 0.941352i $$-0.390444\pi$$
0.337427 + 0.941352i $$0.390444\pi$$
$$24$$ 0 0
$$25$$ 234.406 1.87525
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −232.572 −1.48923 −0.744613 0.667496i $$-0.767366\pi$$
−0.744613 + 0.667496i $$0.767366\pi$$
$$30$$ 0 0
$$31$$ 10.3677 0.0600677 0.0300339 0.999549i $$-0.490438\pi$$
0.0300339 + 0.999549i $$0.490438\pi$$
$$32$$ 0 0
$$33$$ −164.192 −0.866128
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −245.987 −1.09297 −0.546486 0.837468i $$-0.684035\pi$$
−0.546486 + 0.837468i $$0.684035\pi$$
$$38$$ 0 0
$$39$$ −186.052 −0.763901
$$40$$ 0 0
$$41$$ −238.653 −0.909056 −0.454528 0.890733i $$-0.650192\pi$$
−0.454528 + 0.890733i $$0.650192\pi$$
$$42$$ 0 0
$$43$$ 92.9718 0.329722 0.164861 0.986317i $$-0.447282\pi$$
0.164861 + 0.986317i $$0.447282\pi$$
$$44$$ 0 0
$$45$$ −170.622 −0.565219
$$46$$ 0 0
$$47$$ −485.645 −1.50720 −0.753601 0.657332i $$-0.771685\pi$$
−0.753601 + 0.657332i $$0.771685\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −367.317 −1.00852
$$52$$ 0 0
$$53$$ −378.557 −0.981109 −0.490555 0.871410i $$-0.663206\pi$$
−0.490555 + 0.871410i $$0.663206\pi$$
$$54$$ 0 0
$$55$$ 1037.59 2.54379
$$56$$ 0 0
$$57$$ −37.5173 −0.0871804
$$58$$ 0 0
$$59$$ −182.784 −0.403329 −0.201664 0.979455i $$-0.564635\pi$$
−0.201664 + 0.979455i $$0.564635\pi$$
$$60$$ 0 0
$$61$$ −396.470 −0.832177 −0.416088 0.909324i $$-0.636599\pi$$
−0.416088 + 0.909324i $$0.636599\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1175.73 2.24355
$$66$$ 0 0
$$67$$ 261.239 0.476350 0.238175 0.971222i $$-0.423451\pi$$
0.238175 + 0.971222i $$0.423451\pi$$
$$68$$ 0 0
$$69$$ 223.317 0.389627
$$70$$ 0 0
$$71$$ 874.523 1.46179 0.730893 0.682492i $$-0.239104\pi$$
0.730893 + 0.682492i $$0.239104\pi$$
$$72$$ 0 0
$$73$$ −152.406 −0.244354 −0.122177 0.992508i $$-0.538988\pi$$
−0.122177 + 0.992508i $$0.538988\pi$$
$$74$$ 0 0
$$75$$ 703.219 1.08268
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −573.357 −0.816554 −0.408277 0.912858i $$-0.633870\pi$$
−0.408277 + 0.912858i $$0.633870\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 317.754 0.420217 0.210108 0.977678i $$-0.432618\pi$$
0.210108 + 0.977678i $$0.432618\pi$$
$$84$$ 0 0
$$85$$ 2321.20 2.96200
$$86$$ 0 0
$$87$$ −697.717 −0.859806
$$88$$ 0 0
$$89$$ 95.0160 0.113165 0.0565824 0.998398i $$-0.481980\pi$$
0.0565824 + 0.998398i $$0.481980\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 31.1032 0.0346801
$$94$$ 0 0
$$95$$ 237.084 0.256046
$$96$$ 0 0
$$97$$ 1608.78 1.68398 0.841992 0.539490i $$-0.181383\pi$$
0.841992 + 0.539490i $$0.181383\pi$$
$$98$$ 0 0
$$99$$ −492.577 −0.500059
$$100$$ 0 0
$$101$$ 783.043 0.771442 0.385721 0.922615i $$-0.373953\pi$$
0.385721 + 0.922615i $$0.373953\pi$$
$$102$$ 0 0
$$103$$ −1489.61 −1.42501 −0.712503 0.701669i $$-0.752438\pi$$
−0.712503 + 0.701669i $$0.752438\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −712.769 −0.643981 −0.321991 0.946743i $$-0.604352\pi$$
−0.321991 + 0.946743i $$0.604352\pi$$
$$108$$ 0 0
$$109$$ 1041.85 0.915513 0.457757 0.889078i $$-0.348653\pi$$
0.457757 + 0.889078i $$0.348653\pi$$
$$110$$ 0 0
$$111$$ −737.960 −0.631028
$$112$$ 0 0
$$113$$ 352.093 0.293116 0.146558 0.989202i $$-0.453181\pi$$
0.146558 + 0.989202i $$0.453181\pi$$
$$114$$ 0 0
$$115$$ −1411.22 −1.14432
$$116$$ 0 0
$$117$$ −558.156 −0.441039
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1664.46 1.25053
$$122$$ 0 0
$$123$$ −715.958 −0.524844
$$124$$ 0 0
$$125$$ −2074.13 −1.48413
$$126$$ 0 0
$$127$$ 1093.73 0.764194 0.382097 0.924122i $$-0.375202\pi$$
0.382097 + 0.924122i $$0.375202\pi$$
$$128$$ 0 0
$$129$$ 278.915 0.190365
$$130$$ 0 0
$$131$$ −2166.37 −1.44486 −0.722430 0.691444i $$-0.756975\pi$$
−0.722430 + 0.691444i $$0.756975\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −511.866 −0.326329
$$136$$ 0 0
$$137$$ −1964.92 −1.22536 −0.612681 0.790330i $$-0.709909\pi$$
−0.612681 + 0.790330i $$0.709909\pi$$
$$138$$ 0 0
$$139$$ −136.976 −0.0835840 −0.0417920 0.999126i $$-0.513307\pi$$
−0.0417920 + 0.999126i $$0.513307\pi$$
$$140$$ 0 0
$$141$$ −1456.93 −0.870184
$$142$$ 0 0
$$143$$ 3394.26 1.98491
$$144$$ 0 0
$$145$$ 4409.11 2.52522
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −280.105 −0.154007 −0.0770037 0.997031i $$-0.524535\pi$$
−0.0770037 + 0.997031i $$0.524535\pi$$
$$150$$ 0 0
$$151$$ −2194.13 −1.18249 −0.591245 0.806492i $$-0.701363\pi$$
−0.591245 + 0.806492i $$0.701363\pi$$
$$152$$ 0 0
$$153$$ −1101.95 −0.582271
$$154$$ 0 0
$$155$$ −196.552 −0.101854
$$156$$ 0 0
$$157$$ 220.818 0.112250 0.0561248 0.998424i $$-0.482126\pi$$
0.0561248 + 0.998424i $$0.482126\pi$$
$$158$$ 0 0
$$159$$ −1135.67 −0.566444
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1476.86 −0.709674 −0.354837 0.934928i $$-0.615464\pi$$
−0.354837 + 0.934928i $$0.615464\pi$$
$$164$$ 0 0
$$165$$ 3112.76 1.46866
$$166$$ 0 0
$$167$$ 2197.66 1.01832 0.509162 0.860671i $$-0.329955\pi$$
0.509162 + 0.860671i $$0.329955\pi$$
$$168$$ 0 0
$$169$$ 1649.15 0.750635
$$170$$ 0 0
$$171$$ −112.552 −0.0503336
$$172$$ 0 0
$$173$$ −2091.61 −0.919201 −0.459601 0.888126i $$-0.652007\pi$$
−0.459601 + 0.888126i $$0.652007\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −548.351 −0.232862
$$178$$ 0 0
$$179$$ −2334.54 −0.974813 −0.487406 0.873175i $$-0.662057\pi$$
−0.487406 + 0.873175i $$0.662057\pi$$
$$180$$ 0 0
$$181$$ −1758.40 −0.722105 −0.361053 0.932545i $$-0.617583\pi$$
−0.361053 + 0.932545i $$0.617583\pi$$
$$182$$ 0 0
$$183$$ −1189.41 −0.480457
$$184$$ 0 0
$$185$$ 4663.42 1.85330
$$186$$ 0 0
$$187$$ 6701.19 2.62053
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3700.68 1.40195 0.700973 0.713188i $$-0.252749\pi$$
0.700973 + 0.713188i $$0.252749\pi$$
$$192$$ 0 0
$$193$$ 2708.45 1.01015 0.505073 0.863077i $$-0.331466\pi$$
0.505073 + 0.863077i $$0.331466\pi$$
$$194$$ 0 0
$$195$$ 3527.18 1.29531
$$196$$ 0 0
$$197$$ −160.686 −0.0581138 −0.0290569 0.999578i $$-0.509250\pi$$
−0.0290569 + 0.999578i $$0.509250\pi$$
$$198$$ 0 0
$$199$$ −2065.99 −0.735951 −0.367976 0.929835i $$-0.619949\pi$$
−0.367976 + 0.929835i $$0.619949\pi$$
$$200$$ 0 0
$$201$$ 783.717 0.275021
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4524.38 1.54145
$$206$$ 0 0
$$207$$ 669.952 0.224951
$$208$$ 0 0
$$209$$ 684.450 0.226528
$$210$$ 0 0
$$211$$ 1007.12 0.328592 0.164296 0.986411i $$-0.447465\pi$$
0.164296 + 0.986411i $$0.447465\pi$$
$$212$$ 0 0
$$213$$ 2623.57 0.843962
$$214$$ 0 0
$$215$$ −1762.56 −0.559096
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −457.219 −0.141078
$$220$$ 0 0
$$221$$ 7593.34 2.31124
$$222$$ 0 0
$$223$$ 1644.83 0.493929 0.246964 0.969025i $$-0.420567\pi$$
0.246964 + 0.969025i $$0.420567\pi$$
$$224$$ 0 0
$$225$$ 2109.66 0.625084
$$226$$ 0 0
$$227$$ 318.874 0.0932354 0.0466177 0.998913i $$-0.485156\pi$$
0.0466177 + 0.998913i $$0.485156\pi$$
$$228$$ 0 0
$$229$$ −2536.11 −0.731837 −0.365918 0.930647i $$-0.619245\pi$$
−0.365918 + 0.930647i $$0.619245\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2332.81 0.655913 0.327957 0.944693i $$-0.393640\pi$$
0.327957 + 0.944693i $$0.393640\pi$$
$$234$$ 0 0
$$235$$ 9206.86 2.55570
$$236$$ 0 0
$$237$$ −1720.07 −0.471438
$$238$$ 0 0
$$239$$ −2713.85 −0.734495 −0.367248 0.930123i $$-0.619700\pi$$
−0.367248 + 0.930123i $$0.619700\pi$$
$$240$$ 0 0
$$241$$ 4174.59 1.11580 0.557902 0.829907i $$-0.311606\pi$$
0.557902 + 0.829907i $$0.311606\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 775.573 0.199792
$$248$$ 0 0
$$249$$ 953.261 0.242612
$$250$$ 0 0
$$251$$ 6123.58 1.53991 0.769954 0.638099i $$-0.220279\pi$$
0.769954 + 0.638099i $$0.220279\pi$$
$$252$$ 0 0
$$253$$ −4074.11 −1.01240
$$254$$ 0 0
$$255$$ 6963.61 1.71011
$$256$$ 0 0
$$257$$ −4633.91 −1.12473 −0.562365 0.826889i $$-0.690108\pi$$
−0.562365 + 0.826889i $$0.690108\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2093.15 −0.496409
$$262$$ 0 0
$$263$$ −3283.23 −0.769781 −0.384891 0.922962i $$-0.625761\pi$$
−0.384891 + 0.922962i $$0.625761\pi$$
$$264$$ 0 0
$$265$$ 7176.69 1.66362
$$266$$ 0 0
$$267$$ 285.048 0.0653358
$$268$$ 0 0
$$269$$ 798.985 0.181097 0.0905483 0.995892i $$-0.471138\pi$$
0.0905483 + 0.995892i $$0.471138\pi$$
$$270$$ 0 0
$$271$$ −6706.07 −1.50319 −0.751596 0.659624i $$-0.770716\pi$$
−0.751596 + 0.659624i $$0.770716\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12829.3 −2.81321
$$276$$ 0 0
$$277$$ 1162.27 0.252108 0.126054 0.992023i $$-0.459769\pi$$
0.126054 + 0.992023i $$0.459769\pi$$
$$278$$ 0 0
$$279$$ 93.3096 0.0200226
$$280$$ 0 0
$$281$$ 2718.17 0.577055 0.288527 0.957472i $$-0.406834\pi$$
0.288527 + 0.957472i $$0.406834\pi$$
$$282$$ 0 0
$$283$$ −3009.48 −0.632138 −0.316069 0.948736i $$-0.602363\pi$$
−0.316069 + 0.948736i $$0.602363\pi$$
$$284$$ 0 0
$$285$$ 711.253 0.147828
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 10078.3 2.05136
$$290$$ 0 0
$$291$$ 4826.33 0.972249
$$292$$ 0 0
$$293$$ −4209.15 −0.839252 −0.419626 0.907697i $$-0.637839\pi$$
−0.419626 + 0.907697i $$0.637839\pi$$
$$294$$ 0 0
$$295$$ 3465.21 0.683907
$$296$$ 0 0
$$297$$ −1477.73 −0.288709
$$298$$ 0 0
$$299$$ −4616.51 −0.892909
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2349.13 0.445392
$$304$$ 0 0
$$305$$ 7516.28 1.41109
$$306$$ 0 0
$$307$$ −3114.82 −0.579063 −0.289531 0.957169i $$-0.593499\pi$$
−0.289531 + 0.957169i $$0.593499\pi$$
$$308$$ 0 0
$$309$$ −4468.83 −0.822727
$$310$$ 0 0
$$311$$ −8487.24 −1.54748 −0.773741 0.633501i $$-0.781617\pi$$
−0.773741 + 0.633501i $$0.781617\pi$$
$$312$$ 0 0
$$313$$ 4954.91 0.894786 0.447393 0.894337i $$-0.352352\pi$$
0.447393 + 0.894337i $$0.352352\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5608.79 −0.993757 −0.496879 0.867820i $$-0.665521\pi$$
−0.496879 + 0.867820i $$0.665521\pi$$
$$318$$ 0 0
$$319$$ 12728.9 2.23411
$$320$$ 0 0
$$321$$ −2138.31 −0.371803
$$322$$ 0 0
$$323$$ 1531.19 0.263770
$$324$$ 0 0
$$325$$ −14537.3 −2.48117
$$326$$ 0 0
$$327$$ 3125.54 0.528572
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 6609.49 1.09755 0.548777 0.835969i $$-0.315094\pi$$
0.548777 + 0.835969i $$0.315094\pi$$
$$332$$ 0 0
$$333$$ −2213.88 −0.364324
$$334$$ 0 0
$$335$$ −4952.57 −0.807725
$$336$$ 0 0
$$337$$ 11455.5 1.85170 0.925850 0.377891i $$-0.123350\pi$$
0.925850 + 0.377891i $$0.123350\pi$$
$$338$$ 0 0
$$339$$ 1056.28 0.169231
$$340$$ 0 0
$$341$$ −567.434 −0.0901123
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4233.65 −0.660673
$$346$$ 0 0
$$347$$ 6044.15 0.935064 0.467532 0.883976i $$-0.345143\pi$$
0.467532 + 0.883976i $$0.345143\pi$$
$$348$$ 0 0
$$349$$ 5487.93 0.841726 0.420863 0.907124i $$-0.361727\pi$$
0.420863 + 0.907124i $$0.361727\pi$$
$$350$$ 0 0
$$351$$ −1674.47 −0.254634
$$352$$ 0 0
$$353$$ −6880.15 −1.03738 −0.518688 0.854964i $$-0.673579\pi$$
−0.518688 + 0.854964i $$0.673579\pi$$
$$354$$ 0 0
$$355$$ −16579.2 −2.47869
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3938.34 0.578990 0.289495 0.957180i $$-0.406513\pi$$
0.289495 + 0.957180i $$0.406513\pi$$
$$360$$ 0 0
$$361$$ −6702.61 −0.977199
$$362$$ 0 0
$$363$$ 4993.38 0.721996
$$364$$ 0 0
$$365$$ 2889.32 0.414340
$$366$$ 0 0
$$367$$ 1439.63 0.204763 0.102381 0.994745i $$-0.467354\pi$$
0.102381 + 0.994745i $$0.467354\pi$$
$$368$$ 0 0
$$369$$ −2147.87 −0.303019
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2494.11 −0.346220 −0.173110 0.984903i $$-0.555382\pi$$
−0.173110 + 0.984903i $$0.555382\pi$$
$$374$$ 0 0
$$375$$ −6222.39 −0.856861
$$376$$ 0 0
$$377$$ 14423.5 1.97042
$$378$$ 0 0
$$379$$ 1309.25 0.177445 0.0887225 0.996056i $$-0.471722\pi$$
0.0887225 + 0.996056i $$0.471722\pi$$
$$380$$ 0 0
$$381$$ 3281.18 0.441208
$$382$$ 0 0
$$383$$ 6961.80 0.928803 0.464402 0.885625i $$-0.346269\pi$$
0.464402 + 0.885625i $$0.346269\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 836.746 0.109907
$$388$$ 0 0
$$389$$ 6353.45 0.828104 0.414052 0.910253i $$-0.364113\pi$$
0.414052 + 0.910253i $$0.364113\pi$$
$$390$$ 0 0
$$391$$ −9114.25 −1.17884
$$392$$ 0 0
$$393$$ −6499.11 −0.834191
$$394$$ 0 0
$$395$$ 10869.7 1.38459
$$396$$ 0 0
$$397$$ −11505.4 −1.45451 −0.727254 0.686368i $$-0.759204\pi$$
−0.727254 + 0.686368i $$0.759204\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1653.33 0.205893 0.102947 0.994687i $$-0.467173\pi$$
0.102947 + 0.994687i $$0.467173\pi$$
$$402$$ 0 0
$$403$$ −642.979 −0.0794766
$$404$$ 0 0
$$405$$ −1535.60 −0.188406
$$406$$ 0 0
$$407$$ 13463.0 1.63965
$$408$$ 0 0
$$409$$ 4894.83 0.591769 0.295885 0.955224i $$-0.404386\pi$$
0.295885 + 0.955224i $$0.404386\pi$$
$$410$$ 0 0
$$411$$ −5894.77 −0.707463
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −6023.98 −0.712544
$$416$$ 0 0
$$417$$ −410.929 −0.0482573
$$418$$ 0 0
$$419$$ −265.504 −0.0309563 −0.0154782 0.999880i $$-0.504927\pi$$
−0.0154782 + 0.999880i $$0.504927\pi$$
$$420$$ 0 0
$$421$$ 11136.8 1.28925 0.644623 0.764500i $$-0.277014\pi$$
0.644623 + 0.764500i $$0.277014\pi$$
$$422$$ 0 0
$$423$$ −4370.80 −0.502401
$$424$$ 0 0
$$425$$ −28700.5 −3.27572
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 10182.8 1.14599
$$430$$ 0 0
$$431$$ −5193.41 −0.580413 −0.290206 0.956964i $$-0.593724\pi$$
−0.290206 + 0.956964i $$0.593724\pi$$
$$432$$ 0 0
$$433$$ −6314.17 −0.700785 −0.350392 0.936603i $$-0.613952\pi$$
−0.350392 + 0.936603i $$0.613952\pi$$
$$434$$ 0 0
$$435$$ 13227.3 1.45794
$$436$$ 0 0
$$437$$ −930.917 −0.101903
$$438$$ 0 0
$$439$$ −15423.3 −1.67680 −0.838399 0.545056i $$-0.816508\pi$$
−0.838399 + 0.545056i $$0.816508\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 8706.11 0.933725 0.466862 0.884330i $$-0.345384\pi$$
0.466862 + 0.884330i $$0.345384\pi$$
$$444$$ 0 0
$$445$$ −1801.31 −0.191889
$$446$$ 0 0
$$447$$ −840.315 −0.0889162
$$448$$ 0 0
$$449$$ 5495.91 0.577657 0.288829 0.957381i $$-0.406734\pi$$
0.288829 + 0.957381i $$0.406734\pi$$
$$450$$ 0 0
$$451$$ 13061.7 1.36375
$$452$$ 0 0
$$453$$ −6582.40 −0.682711
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11357.3 −1.16252 −0.581258 0.813719i $$-0.697439\pi$$
−0.581258 + 0.813719i $$0.697439\pi$$
$$458$$ 0 0
$$459$$ −3305.86 −0.336175
$$460$$ 0 0
$$461$$ −14514.2 −1.46637 −0.733184 0.680030i $$-0.761967\pi$$
−0.733184 + 0.680030i $$0.761967\pi$$
$$462$$ 0 0
$$463$$ −9971.00 −1.00085 −0.500423 0.865781i $$-0.666822\pi$$
−0.500423 + 0.865781i $$0.666822\pi$$
$$464$$ 0 0
$$465$$ −589.655 −0.0588056
$$466$$ 0 0
$$467$$ −397.477 −0.0393856 −0.0196928 0.999806i $$-0.506269\pi$$
−0.0196928 + 0.999806i $$0.506269\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 662.454 0.0648074
$$472$$ 0 0
$$473$$ −5088.42 −0.494642
$$474$$ 0 0
$$475$$ −2931.43 −0.283165
$$476$$ 0 0
$$477$$ −3407.01 −0.327036
$$478$$ 0 0
$$479$$ −14060.4 −1.34120 −0.670602 0.741818i $$-0.733964\pi$$
−0.670602 + 0.741818i $$0.733964\pi$$
$$480$$ 0 0
$$481$$ 15255.4 1.44613
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −30499.2 −2.85546
$$486$$ 0 0
$$487$$ 13534.7 1.25937 0.629687 0.776849i $$-0.283183\pi$$
0.629687 + 0.776849i $$0.283183\pi$$
$$488$$ 0 0
$$489$$ −4430.59 −0.409730
$$490$$ 0 0
$$491$$ 8693.29 0.799028 0.399514 0.916727i $$-0.369179\pi$$
0.399514 + 0.916727i $$0.369179\pi$$
$$492$$ 0 0
$$493$$ 28475.9 2.60140
$$494$$ 0 0
$$495$$ 9338.29 0.847929
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4017.93 −0.360455 −0.180228 0.983625i $$-0.557683\pi$$
−0.180228 + 0.983625i $$0.557683\pi$$
$$500$$ 0 0
$$501$$ 6592.98 0.587929
$$502$$ 0 0
$$503$$ 52.2455 0.00463124 0.00231562 0.999997i $$-0.499263\pi$$
0.00231562 + 0.999997i $$0.499263\pi$$
$$504$$ 0 0
$$505$$ −14844.9 −1.30810
$$506$$ 0 0
$$507$$ 4947.44 0.433379
$$508$$ 0 0
$$509$$ −9239.10 −0.804550 −0.402275 0.915519i $$-0.631780\pi$$
−0.402275 + 0.915519i $$0.631780\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −337.655 −0.0290601
$$514$$ 0 0
$$515$$ 28240.0 2.41632
$$516$$ 0 0
$$517$$ 26579.7 2.26107
$$518$$ 0 0
$$519$$ −6274.82 −0.530701
$$520$$ 0 0
$$521$$ −14973.4 −1.25911 −0.629555 0.776956i $$-0.716763\pi$$
−0.629555 + 0.776956i $$0.716763\pi$$
$$522$$ 0 0
$$523$$ 13603.5 1.13736 0.568681 0.822558i $$-0.307454\pi$$
0.568681 + 0.822558i $$0.307454\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1269.42 −0.104927
$$528$$ 0 0
$$529$$ −6625.82 −0.544573
$$530$$ 0 0
$$531$$ −1645.05 −0.134443
$$532$$ 0 0
$$533$$ 14800.6 1.20279
$$534$$ 0 0
$$535$$ 13512.7 1.09197
$$536$$ 0 0
$$537$$ −7003.61 −0.562808
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16886.4 −1.34196 −0.670981 0.741474i $$-0.734127\pi$$
−0.670981 + 0.741474i $$0.734127\pi$$
$$542$$ 0 0
$$543$$ −5275.21 −0.416908
$$544$$ 0 0
$$545$$ −19751.4 −1.55240
$$546$$ 0 0
$$547$$ 5987.25 0.468001 0.234000 0.972237i $$-0.424818\pi$$
0.234000 + 0.972237i $$0.424818\pi$$
$$548$$ 0 0
$$549$$ −3568.23 −0.277392
$$550$$ 0 0
$$551$$ 2908.49 0.224875
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 13990.3 1.07001
$$556$$ 0 0
$$557$$ 2238.71 0.170300 0.0851499 0.996368i $$-0.472863\pi$$
0.0851499 + 0.996368i $$0.472863\pi$$
$$558$$ 0 0
$$559$$ −5765.86 −0.436261
$$560$$ 0 0
$$561$$ 20103.6 1.51297
$$562$$ 0 0
$$563$$ 8452.84 0.632762 0.316381 0.948632i $$-0.397532\pi$$
0.316381 + 0.948632i $$0.397532\pi$$
$$564$$ 0 0
$$565$$ −6674.98 −0.497024
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 14953.6 1.10173 0.550866 0.834594i $$-0.314298\pi$$
0.550866 + 0.834594i $$0.314298\pi$$
$$570$$ 0 0
$$571$$ −16021.8 −1.17424 −0.587122 0.809498i $$-0.699739\pi$$
−0.587122 + 0.809498i $$0.699739\pi$$
$$572$$ 0 0
$$573$$ 11102.0 0.809414
$$574$$ 0 0
$$575$$ 17449.0 1.26552
$$576$$ 0 0
$$577$$ 16113.1 1.16256 0.581279 0.813704i $$-0.302552\pi$$
0.581279 + 0.813704i $$0.302552\pi$$
$$578$$ 0 0
$$579$$ 8125.34 0.583208
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 20718.7 1.47184
$$584$$ 0 0
$$585$$ 10581.5 0.747850
$$586$$ 0 0
$$587$$ −9552.04 −0.671644 −0.335822 0.941925i $$-0.609014\pi$$
−0.335822 + 0.941925i $$0.609014\pi$$
$$588$$ 0 0
$$589$$ −129.656 −0.00907028
$$590$$ 0 0
$$591$$ −482.058 −0.0335520
$$592$$ 0 0
$$593$$ 24167.8 1.67361 0.836807 0.547499i $$-0.184420\pi$$
0.836807 + 0.547499i $$0.184420\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6197.98 −0.424902
$$598$$ 0 0
$$599$$ −821.331 −0.0560245 −0.0280122 0.999608i $$-0.508918\pi$$
−0.0280122 + 0.999608i $$0.508918\pi$$
$$600$$ 0 0
$$601$$ 14674.7 0.995996 0.497998 0.867178i $$-0.334069\pi$$
0.497998 + 0.867178i $$0.334069\pi$$
$$602$$ 0 0
$$603$$ 2351.15 0.158783
$$604$$ 0 0
$$605$$ −31554.9 −2.12048
$$606$$ 0 0
$$607$$ 5083.12 0.339897 0.169949 0.985453i $$-0.445640\pi$$
0.169949 + 0.985453i $$0.445640\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 30118.4 1.99420
$$612$$ 0 0
$$613$$ −22202.0 −1.46286 −0.731428 0.681919i $$-0.761146\pi$$
−0.731428 + 0.681919i $$0.761146\pi$$
$$614$$ 0 0
$$615$$ 13573.1 0.889954
$$616$$ 0 0
$$617$$ −14990.6 −0.978121 −0.489060 0.872250i $$-0.662660\pi$$
−0.489060 + 0.872250i $$0.662660\pi$$
$$618$$ 0 0
$$619$$ 3073.02 0.199540 0.0997700 0.995011i $$-0.468189\pi$$
0.0997700 + 0.995011i $$0.468189\pi$$
$$620$$ 0 0
$$621$$ 2009.86 0.129876
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 10020.6 0.641317
$$626$$ 0 0
$$627$$ 2053.35 0.130786
$$628$$ 0 0
$$629$$ 30118.4 1.90922
$$630$$ 0 0
$$631$$ −26012.7 −1.64113 −0.820563 0.571556i $$-0.806340\pi$$
−0.820563 + 0.571556i $$0.806340\pi$$
$$632$$ 0 0
$$633$$ 3021.36 0.189713
$$634$$ 0 0
$$635$$ −20734.9 −1.29581
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 7870.70 0.487262
$$640$$ 0 0
$$641$$ 8032.23 0.494936 0.247468 0.968896i $$-0.420401\pi$$
0.247468 + 0.968896i $$0.420401\pi$$
$$642$$ 0 0
$$643$$ −24887.7 −1.52640 −0.763200 0.646162i $$-0.776373\pi$$
−0.763200 + 0.646162i $$0.776373\pi$$
$$644$$ 0 0
$$645$$ −5287.68 −0.322794
$$646$$ 0 0
$$647$$ −21084.3 −1.28116 −0.640580 0.767891i $$-0.721306\pi$$
−0.640580 + 0.767891i $$0.721306\pi$$
$$648$$ 0 0
$$649$$ 10003.9 0.605065
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 13166.3 0.789033 0.394516 0.918889i $$-0.370912\pi$$
0.394516 + 0.918889i $$0.370912\pi$$
$$654$$ 0 0
$$655$$ 41070.1 2.44999
$$656$$ 0 0
$$657$$ −1371.66 −0.0814513
$$658$$ 0 0
$$659$$ 13903.4 0.821851 0.410926 0.911669i $$-0.365206\pi$$
0.410926 + 0.911669i $$0.365206\pi$$
$$660$$ 0 0
$$661$$ −6306.61 −0.371102 −0.185551 0.982635i $$-0.559407\pi$$
−0.185551 + 0.982635i $$0.559407\pi$$
$$662$$ 0 0
$$663$$ 22780.0 1.33439
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −17312.5 −1.00501
$$668$$ 0 0
$$669$$ 4934.50 0.285170
$$670$$ 0 0
$$671$$ 21699.1 1.24841
$$672$$ 0 0
$$673$$ −24407.6 −1.39798 −0.698992 0.715129i $$-0.746368\pi$$
−0.698992 + 0.715129i $$0.746368\pi$$
$$674$$ 0 0
$$675$$ 6328.97 0.360892
$$676$$ 0 0
$$677$$ 31080.3 1.76442 0.882209 0.470858i $$-0.156055\pi$$
0.882209 + 0.470858i $$0.156055\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 956.623 0.0538295
$$682$$ 0 0
$$683$$ −28759.5 −1.61120 −0.805601 0.592458i $$-0.798157\pi$$
−0.805601 + 0.592458i $$0.798157\pi$$
$$684$$ 0 0
$$685$$ 37251.0 2.07779
$$686$$ 0 0
$$687$$ −7608.32 −0.422526
$$688$$ 0 0
$$689$$ 23477.1 1.29812
$$690$$ 0 0
$$691$$ 24895.4 1.37057 0.685287 0.728273i $$-0.259677\pi$$
0.685287 + 0.728273i $$0.259677\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2596.80 0.141730
$$696$$ 0 0
$$697$$ 29220.4 1.58795
$$698$$ 0 0
$$699$$ 6998.44 0.378692
$$700$$ 0 0
$$701$$ −1702.74 −0.0917427 −0.0458714 0.998947i $$-0.514606\pi$$
−0.0458714 + 0.998947i $$0.514606\pi$$
$$702$$ 0 0
$$703$$ 3076.25 0.165040
$$704$$ 0 0
$$705$$ 27620.6 1.47553
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6523.21 0.345535 0.172767 0.984963i $$-0.444729\pi$$
0.172767 + 0.984963i $$0.444729\pi$$
$$710$$ 0 0
$$711$$ −5160.22 −0.272185
$$712$$ 0 0
$$713$$ 771.765 0.0405369
$$714$$ 0 0
$$715$$ −64348.4 −3.36572
$$716$$ 0 0
$$717$$ −8141.55 −0.424061
$$718$$ 0 0
$$719$$ −25254.4 −1.30992 −0.654959 0.755664i $$-0.727314\pi$$
−0.654959 + 0.755664i $$0.727314\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 12523.8 0.644210
$$724$$ 0 0
$$725$$ −54516.4 −2.79268
$$726$$ 0 0
$$727$$ −27964.9 −1.42663 −0.713316 0.700843i $$-0.752808\pi$$
−0.713316 + 0.700843i $$0.752808\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −11383.4 −0.575964
$$732$$ 0 0
$$733$$ 9294.38 0.468344 0.234172 0.972195i $$-0.424762\pi$$
0.234172 + 0.972195i $$0.424762\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −14297.8 −0.714609
$$738$$ 0 0
$$739$$ −18291.2 −0.910490 −0.455245 0.890366i $$-0.650448\pi$$
−0.455245 + 0.890366i $$0.650448\pi$$
$$740$$ 0 0
$$741$$ 2326.72 0.115350
$$742$$ 0 0
$$743$$ −14742.4 −0.727921 −0.363960 0.931414i $$-0.618576\pi$$
−0.363960 + 0.931414i $$0.618576\pi$$
$$744$$ 0 0
$$745$$ 5310.24 0.261144
$$746$$ 0 0
$$747$$ 2859.78 0.140072
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −28463.4 −1.38301 −0.691507 0.722370i $$-0.743053\pi$$
−0.691507 + 0.722370i $$0.743053\pi$$
$$752$$ 0 0
$$753$$ 18370.7 0.889067
$$754$$ 0 0
$$755$$ 41596.4 2.00510
$$756$$ 0 0
$$757$$ −20336.7 −0.976422 −0.488211 0.872726i $$-0.662350\pi$$
−0.488211 + 0.872726i $$0.662350\pi$$
$$758$$ 0 0
$$759$$ −12222.3 −0.584509
$$760$$ 0 0
$$761$$ −39581.6 −1.88546 −0.942728 0.333564i $$-0.891749\pi$$
−0.942728 + 0.333564i $$0.891749\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 20890.8 0.987332
$$766$$ 0 0
$$767$$ 11335.7 0.533650
$$768$$ 0 0
$$769$$ −10580.3 −0.496146 −0.248073 0.968741i $$-0.579797\pi$$
−0.248073 + 0.968741i $$0.579797\pi$$
$$770$$ 0 0
$$771$$ −13901.7 −0.649363
$$772$$ 0 0
$$773$$ −25922.2 −1.20615 −0.603077 0.797683i $$-0.706059\pi$$
−0.603077 + 0.797683i $$0.706059\pi$$
$$774$$ 0 0
$$775$$ 2430.26 0.112642
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2984.53 0.137268
$$780$$ 0 0
$$781$$ −47863.3 −2.19294
$$782$$ 0 0
$$783$$ −6279.45 −0.286602
$$784$$ 0 0
$$785$$ −4186.27 −0.190337
$$786$$ 0 0
$$787$$ 17220.0 0.779958 0.389979 0.920824i $$-0.372482\pi$$
0.389979 + 0.920824i $$0.372482\pi$$
$$788$$ 0 0
$$789$$ −9849.68 −0.444433
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 24588.0 1.10107
$$794$$ 0 0
$$795$$ 21530.1 0.960494
$$796$$ 0 0
$$797$$ 6275.52 0.278909 0.139454 0.990228i $$-0.455465\pi$$
0.139454 + 0.990228i $$0.455465\pi$$
$$798$$ 0 0
$$799$$ 59461.9 2.63280
$$800$$ 0 0
$$801$$ 855.144 0.0377216
$$802$$ 0 0
$$803$$ 8341.33 0.366574
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 2396.96 0.104556
$$808$$ 0 0
$$809$$ −4604.03 −0.200085 −0.100043 0.994983i $$-0.531898\pi$$
−0.100043 + 0.994983i $$0.531898\pi$$
$$810$$ 0 0
$$811$$ 5104.36 0.221009 0.110505 0.993876i $$-0.464753\pi$$
0.110505 + 0.993876i $$0.464753\pi$$
$$812$$ 0 0
$$813$$ −20118.2 −0.867868
$$814$$ 0 0
$$815$$ 27998.4 1.20336
$$816$$ 0 0
$$817$$ −1162.68 −0.0497884
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −45635.4 −1.93993 −0.969967 0.243237i $$-0.921791\pi$$
−0.969967 + 0.243237i $$0.921791\pi$$
$$822$$ 0 0
$$823$$ 39377.1 1.66780 0.833901 0.551915i $$-0.186103\pi$$
0.833901 + 0.551915i $$0.186103\pi$$
$$824$$ 0 0
$$825$$ −38487.8 −1.62421
$$826$$ 0 0
$$827$$ 30916.3 1.29996 0.649978 0.759953i $$-0.274778\pi$$
0.649978 + 0.759953i $$0.274778\pi$$
$$828$$ 0 0
$$829$$ 2543.33 0.106554 0.0532771 0.998580i $$-0.483033\pi$$
0.0532771 + 0.998580i $$0.483033\pi$$
$$830$$ 0 0
$$831$$ 3486.81 0.145555
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −41663.3 −1.72673
$$836$$ 0 0
$$837$$ 279.929 0.0115600
$$838$$ 0 0
$$839$$ −7930.82 −0.326343 −0.163172 0.986598i $$-0.552172\pi$$
−0.163172 + 0.986598i $$0.552172\pi$$
$$840$$ 0 0
$$841$$ 29700.8 1.21780
$$842$$ 0 0
$$843$$ 8154.51 0.333163
$$844$$ 0 0
$$845$$ −31264.5 −1.27282
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9028.45 −0.364965
$$850$$ 0 0
$$851$$ −18311.0 −0.737595
$$852$$ 0 0
$$853$$ −41983.2 −1.68520 −0.842601 0.538538i $$-0.818977\pi$$
−0.842601 + 0.538538i $$0.818977\pi$$
$$854$$ 0 0
$$855$$ 2133.76 0.0853485
$$856$$ 0 0
$$857$$ 11854.3 0.472504 0.236252 0.971692i $$-0.424081\pi$$
0.236252 + 0.971692i $$0.424081\pi$$
$$858$$ 0 0
$$859$$ −8113.88 −0.322284 −0.161142 0.986931i $$-0.551518\pi$$
−0.161142 + 0.986931i $$0.551518\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −45817.0 −1.80722 −0.903609 0.428358i $$-0.859092\pi$$
−0.903609 + 0.428358i $$0.859092\pi$$
$$864$$ 0 0
$$865$$ 39652.7 1.55865
$$866$$ 0 0
$$867$$ 30235.0 1.18435
$$868$$ 0 0
$$869$$ 31380.3 1.22498
$$870$$ 0 0
$$871$$ −16201.3 −0.630266
$$872$$ 0 0
$$873$$ 14479.0 0.561328
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31813.0 1.22491 0.612456 0.790504i $$-0.290182\pi$$
0.612456 + 0.790504i $$0.290182\pi$$
$$878$$ 0 0
$$879$$ −12627.4 −0.484543
$$880$$ 0 0
$$881$$ −43551.6 −1.66548 −0.832742 0.553661i $$-0.813230\pi$$
−0.832742 + 0.553661i $$0.813230\pi$$
$$882$$ 0 0
$$883$$ −40645.1 −1.54906 −0.774528 0.632540i $$-0.782012\pi$$
−0.774528 + 0.632540i $$0.782012\pi$$
$$884$$ 0 0
$$885$$ 10395.6 0.394854
$$886$$ 0 0
$$887$$ 34028.2 1.28811 0.644056 0.764978i $$-0.277250\pi$$
0.644056 + 0.764978i $$0.277250\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4433.20 −0.166686
$$892$$ 0 0
$$893$$ 6073.35 0.227589
$$894$$ 0 0
$$895$$ 44258.2 1.65295
$$896$$ 0 0
$$897$$ −13849.5 −0.515521
$$898$$ 0 0
$$899$$ −2411.25 −0.0894545
$$900$$ 0 0
$$901$$ 46350.2 1.71382
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 33335.8 1.22444
$$906$$ 0 0
$$907$$ −50804.7 −1.85992 −0.929958 0.367666i $$-0.880157\pi$$
−0.929958 + 0.367666i $$0.880157\pi$$
$$908$$ 0 0
$$909$$ 7047.38 0.257147
$$910$$ 0 0
$$911$$ −28738.3 −1.04516 −0.522582 0.852589i $$-0.675031\pi$$
−0.522582 + 0.852589i $$0.675031\pi$$
$$912$$ 0 0
$$913$$ −17390.9 −0.630400
$$914$$ 0 0
$$915$$ 22548.9 0.814691
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 54742.2 1.96494 0.982470 0.186420i $$-0.0596886\pi$$
0.982470 + 0.186420i $$0.0596886\pi$$
$$920$$ 0 0
$$921$$ −9344.46 −0.334322
$$922$$ 0 0
$$923$$ −54235.5 −1.93411
$$924$$ 0 0
$$925$$ −57660.9 −2.04960
$$926$$ 0 0
$$927$$ −13406.5 −0.475002
$$928$$ 0 0
$$929$$ −43775.8 −1.54600 −0.773002 0.634403i $$-0.781246\pi$$
−0.773002 + 0.634403i $$0.781246\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −25461.7 −0.893440
$$934$$ 0 0
$$935$$ −127041. −4.44352
$$936$$ 0 0
$$937$$ −35090.9 −1.22345 −0.611724 0.791071i $$-0.709524\pi$$
−0.611724 + 0.791071i $$0.709524\pi$$
$$938$$ 0 0
$$939$$ 14864.7 0.516605
$$940$$ 0 0
$$941$$ 31784.8 1.10112 0.550560 0.834795i $$-0.314414\pi$$
0.550560 + 0.834795i $$0.314414\pi$$
$$942$$ 0 0
$$943$$ −17765.1 −0.613479
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −56500.9 −1.93879 −0.969394 0.245510i $$-0.921045\pi$$
−0.969394 + 0.245510i $$0.921045\pi$$
$$948$$ 0 0
$$949$$ 9451.84 0.323308
$$950$$ 0 0
$$951$$ −16826.4 −0.573746
$$952$$ 0 0
$$953$$ 36669.1 1.24641 0.623204 0.782059i $$-0.285831\pi$$
0.623204 + 0.782059i $$0.285831\pi$$
$$954$$ 0 0
$$955$$ −70157.5 −2.37722
$$956$$ 0 0
$$957$$ 38186.6 1.28986
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29683.5 −0.996392
$$962$$ 0 0
$$963$$ −6414.92 −0.214660
$$964$$ 0 0
$$965$$ −51346.8 −1.71286
$$966$$ 0 0
$$967$$ 12012.4 0.399474 0.199737 0.979850i $$-0.435991\pi$$
0.199737 + 0.979850i $$0.435991\pi$$
$$968$$ 0 0
$$969$$ 4593.58 0.152288
$$970$$ 0 0
$$971$$ −37296.2 −1.23264 −0.616320 0.787496i $$-0.711377\pi$$
−0.616320 + 0.787496i $$0.711377\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −43611.8 −1.43251
$$976$$ 0 0
$$977$$ 9788.45 0.320532 0.160266 0.987074i $$-0.448765\pi$$
0.160266 + 0.987074i $$0.448765\pi$$
$$978$$ 0 0
$$979$$ −5200.30 −0.169767
$$980$$ 0 0
$$981$$ 9376.63 0.305171
$$982$$ 0 0
$$983$$ −4620.40 −0.149917 −0.0749583 0.997187i $$-0.523882\pi$$
−0.0749583 + 0.997187i $$0.523882\pi$$
$$984$$ 0 0
$$985$$ 3046.29 0.0985410
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 6920.73 0.222514
$$990$$ 0 0
$$991$$ −40365.3 −1.29389 −0.646946 0.762536i $$-0.723954\pi$$
−0.646946 + 0.762536i $$0.723954\pi$$
$$992$$ 0 0
$$993$$ 19828.5 0.633673
$$994$$ 0 0
$$995$$ 39167.1 1.24792
$$996$$ 0 0
$$997$$ 2994.35 0.0951175 0.0475587 0.998868i $$-0.484856\pi$$
0.0475587 + 0.998868i $$0.484856\pi$$
$$998$$ 0 0
$$999$$ −6641.64 −0.210343
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2352.4.a.cp.1.1 4
4.3 odd 2 1176.4.a.ba.1.1 4
7.3 odd 6 336.4.q.m.289.1 8
7.5 odd 6 336.4.q.m.193.1 8
7.6 odd 2 2352.4.a.cm.1.4 4
28.3 even 6 168.4.q.f.121.1 yes 8
28.19 even 6 168.4.q.f.25.1 8
28.27 even 2 1176.4.a.bd.1.4 4
84.47 odd 6 504.4.s.j.361.4 8
84.59 odd 6 504.4.s.j.289.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.1 8 28.19 even 6
168.4.q.f.121.1 yes 8 28.3 even 6
336.4.q.m.193.1 8 7.5 odd 6
336.4.q.m.289.1 8 7.3 odd 6
504.4.s.j.289.4 8 84.59 odd 6
504.4.s.j.361.4 8 84.47 odd 6
1176.4.a.ba.1.1 4 4.3 odd 2
1176.4.a.bd.1.4 4 28.27 even 2
2352.4.a.cm.1.4 4 7.6 odd 2
2352.4.a.cp.1.1 4 1.1 even 1 trivial